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Theorem sucexeloni 7749
Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7751 does not require ax-un 7677. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)
Assertion
Ref Expression
sucexeloni ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloni
StepHypRef Expression
1 eloni 6332 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordsuci 7748 . . 3 (Ord 𝐴 → Ord suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ On → Ord suc 𝐴)
4 elex 3466 . 2 (suc 𝐴𝑉 → suc 𝐴 ∈ V)
5 elong 6330 . . 3 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
65biimparc 481 . 2 ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On)
73, 4, 6syl2an 597 1 ((𝐴 ∈ On ∧ suc 𝐴𝑉) → suc 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3448  Ord word 6321  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  onsuc  7751  1on  8429  2on  8431
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